Within this thesis, we will elaborate on both the approaches that we have described so far, and both analyses will present original contributions of this work. In the first part, we will work on the EFT approach to the binary two-body problem, and we will develop a computational algorithm to compute the Hereditary terms using Feynman diagrams up to 5PN orderThis study was done in collaboration with Raj Patil., using the multi-loop technology to evaluate the complicated two-loop integrals appearing, and we will compare our independent results with the ones appearing in literature [1]}, [2]}, [3]}. Then, after understanding that the correct approach to deal with radiation problems is the CTP formalism, we will repeat this same calculation using Schwinger-Keldysh variables, finding a new way to express In-In amplitudes in terms of Feynman diagrams, and we will recompute the 5PN conservative Lagrangians using this formalism. During the second part of the project instead, we will work in the PM perturbative scheme, and we will design an automatic procedure, fully implemented in Mathematica, to get the one-loop classical and quantum corrections to the scattering angle of a light-like scalar particle under the influence of a massive scalar. Working in d-dimensions within a novel approach, we will find new quantum contributions which were not appearing in previous analyses [4]}, [5]}, [6]}. The appearance of these terms will modify the predictions for the bending angle, and other studies obtained from these works have to be reviewed [7]}, [8]}, [9]} according to our novel results. Eventually, we will adapt this computation to the case of two massive scalar fields, and we will evaluate the 1-loop classical and quantum corrections to the Newtonian potential, finding agreement with the results appearing in literature, proving the universality of the methods implemented.
